AP® Physics 1: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics/units/2/2-6)
Unit 2 · Topic 2.6
2.6 Gravitational Force
Every pair of masses attracts with a gravitational force that weakens with the square of the distance between their centers. Near Earth that force is your weight, mg. Your apparent weight, what a scale reads, is the normal force on you, so it changes whenever you accelerate up or down.
Key terms
- universal gravitation
- gravitational field strength (g)
- weight
- apparent weight
- weightlessness
- inertial vs. gravitational mass
Universal gravitation
Newton's law of universal gravitation: F_g = Gm₁m₂/r², where G = 6.67 × 10⁻¹¹ N·m²/kg² and r is the distance between the two objects' centers of mass, not between their surfaces. The force is always attractive and points along the line joining the centers. Each object pulls on the other with the same size force (third law).
Because of the r² in the bottom, distance has a big effect: doubling r makes the force ¼ as large, and tripling r makes it 1/9. Doubling either mass doubles the force.
Gravitational field
A field describes what a noncontact force would do to an object placed at each point in space. The gravitational field strength is the force per kilogram: g = F_g/m = GM/r², where M is the mass creating the field. Its unit is N/kg, which equals m/s².
If gravity is the only force on an object, its acceleration equals the field strength. At Earth's surface g ≈ 9.8 N/kg, so free-falling objects accelerate at 9.8 m/s². Since the field shrinks as 1/r², at two Earth radii from the center it's only about 2.5 N/kg. At the International Space Station, about 400 km up, it's still roughly 8.7 N/kg, close to 90% of its surface value.
Near a planet's surface the change in r over everyday heights is tiny compared with the planet's radius, so you can treat g as constant.
Weight and apparent weight
Weight is the gravitational force a planet exerts on a nearby object: F_g = mg. Mass (kg) is the same everywhere; weight (N) depends on where you are.
Apparent weight is the size of the normal force on you, which is what a bathroom scale measures. In an elevator with upward acceleration, the floor must push harder than your weight: F_N = m(g + a). With downward acceleration, F_N = m(g − a). Only the direction of the acceleration matters, not the direction of motion.
You feel weightless when the normal force is zero. That happens if gravity is the only force on you, as in free fall or in an orbiting space station, or if no forces act at all. Astronauts in orbit still have a large gravitational force on them; they're weightless because they're falling around Earth.
Equivalence and two kinds of mass
Inertial mass measures how hard it is to change an object's motion (the m in a = F/m). Gravitational mass measures how strongly gravity pulls on it (the m in F = Gm₁m₂/r²). Experiments show these are equal, which is why all objects fall with the same acceleration.
The equivalence principle says an observer in a closed, accelerating room can't tell the difference between the effects of acceleration and gravity. A rocket accelerating at 9.8 m/s² in deep space would feel exactly like standing on Earth.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Field strength on Mars
Mars has a mass of 6.42 × 10²³ kg and a radius of 3.39 × 10⁶ m. Find the gravitational field strength at its surface, and the weight there of a 50 kg rover.
Show the solutionHide the solution
- Step 1: g = GM/r² = (6.67 × 10⁻¹¹)(6.42 × 10²³) ÷ (3.39 × 10⁶)².
- Step 2: Numerator ≈ 4.28 × 10¹³; denominator ≈ 1.15 × 10¹³. g ≈ 3.73 N/kg.
- Step 3: Weight = mg = (50)(3.73) ≈ 186 N, about 38% of its weight on Earth (490 N with g = 9.8 N/kg).
Answer: g ≈ 3.7 N/kg; weight ≈ 190 N
- Example 2Calculator allowed
Scale in an elevator (classic trap)
A 60 kg student stands on a scale in an elevator. Use g = 9.8 m/s². What does the scale read when the elevator (a) moves up at constant speed, (b) moves up while speeding up at 2.0 m/s², (c) moves down while slowing down at 2.0 m/s² and (d) moves down while speeding up at 2.0 m/s²?
Show the solutionHide the solution
- Step 1: Take up as positive. ΣF = F_N − mg = ma, so F_N = m(g + a).
- Step 2: (a) a = 0: F_N = (60)(9.8) = 588 N.
- Step 3: (b) a = +2.0 m/s²: F_N = (60)(11.8) = 708 N.
- Step 4: (c) Moving down but slowing means the acceleration points up: a = +2.0 m/s². F_N = 708 N, the same as (b).
- Step 5: (d) Moving down and speeding up means a = −2.0 m/s²: F_N = (60)(7.8) = 468 N.
- Step 6: The trap is reasoning from the direction of motion. Only the direction of acceleration decides whether the scale reads high or low.
Answer: (a) 588 N; (b) 708 N; (c) 708 N; (d) 468 N
- Example 3Calculator allowed
Scaling a planet
Planet X has twice Earth's mass and twice Earth's radius. How does the gravitational field strength at its surface compare with Earth's?
Show the solutionHide the solution
- Step 1: g = GM/r². Replace M with 2M and r with 2r.
- Step 2: g_X = G(2M)/(2r)² = 2GM/(4r²) = ½ (GM/r²).
- Step 3: So g_X = ½ g_Earth, about 4.9 N/kg if g_Earth = 9.8 N/kg.
Answer: Half of Earth's surface field strength
Common mistakes
- Measuring r from a planet's surface instead of its center. For an orbit, r = planet radius + altitude.
- Thinking astronauts in orbit are weightless because there's no gravity. Gravity is strong there; they're in free fall.
- Using the direction of velocity to decide whether a scale reads more or less than mg. Use the direction of acceleration.
- Confusing mass and weight. Mass is in kg and doesn't change; weight is a force in N and depends on g.
On the exam
- Ratio questions are common: what happens to the force or the field if the distance doubles or a mass triples? Write the formula, substitute the factors and simplify.
- Apparent-weight questions usually show an elevator or a ride. Draw the free-body diagram, pick the acceleration's direction and write F_N − mg = ma.
Connected topics
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Check yourself
4 questions on 2.6 Gravitational Force. Pick an answer to see if you got it, and why.
Two spheres attract each other with a gravitational force F. Each sphere's mass is doubled, and the distance between their centers is also doubled. What is the new force?
A planet has twice Earth's mass and twice Earth's radius. If g at Earth's surface is 10 N/kg, what is g at this planet's surface?
At Earth's surface, g = 9.8 N/kg. What is the gravitational field strength at a height above the surface equal to one Earth radius?
A 60 kg student stands on a bathroom scale in an elevator that accelerates upward at 2.0 m/s². What does the scale read? Use g = 10 m/s².
0 of 4 answered